Finite Group Databasebeta

← #29716659

Subgroups — #29716659

Order 15616.

1072 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840066C211yesmaximal, characteristic, complemented
2C2×#16839862C211yesmaximal, characteristic
3#21451189C211yesmaximal, characteristic, complemented
4#21447154C211yesmaximal, characteristic, complemented
5#21447144C211yesmaximal, characteristic, complemented
6#21446967C211yesmaximal, characteristic, complemented
7#21446957C211yesmaximal, characteristic, complemented
8#21446789C211yesmaximal, characteristic, complemented
9#21446760C211yesmaximal, characteristic, complemented
10#21446153C211yesmaximal, characteristic
11#21446153C211yesmaximal, characteristic
12#21446080C211yesmaximal, characteristic
13#21446066C211yesmaximal, characteristic
14#21445819C211yesmaximal, characteristic
15C2×C61×#554C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18#16840066—24no—
19#16839862—24no—
20C2×#14519710C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×C4⋊#92418C2211yescharacteristic
24C2×#14519700C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30C2×#14519688C2211yescharacteristic
31#16839695—21no—
32#16839695—21no—
33#16839669—21no—
34#16839668C2211yescharacteristic
35#16839668C2211yescharacteristic
36#16839660C2211yescharacteristic
37#16839660C2211yescharacteristic
38#16839658—21no—
39#16839632C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C61×#554—24no—
51C61×#554—24no—
52C2×C4⋊Q8×C61C2211yescharacteristic
53C2×#125×C61C2211yescharacteristic
54C2×#125×C61C2211yescharacteristic
55C2×C22⋊Q8×C61—21no—
56C2×C22⋊Q8×C61—21no—
57C2×C22⋊Q8×C61—21no—
58C2×C4×Q8×C61C2211yescharacteristic
59C2×C4×D4×C61C2211yescharacteristic
60C2×C4×D4×C61C2211yescharacteristic
61C2×C4×D4×C61C2211yescharacteristic
62C22×C4⋊C4×C61—21no—
63C22×Dic122C2311yescharacteristic
64C2×#11758200—22no—
65C2×#11758200—22no—
66C2×#11758200Q814yescomplemented
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C4×Dic122—22no—
72C4×Dic122—22no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×C4×Dic61—21no—
82#14519710—22no—
83#14519710—22no—
84#14519708—22no—
85C4⋊#92418—22no—
86#14519700—22no—
87#14519700—22no—
88#14519700—22no—
89#14519698—22no—
90#14519698—22no—
91#14519698—22no—
92#14519688—22no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99C22×Q8×C61—21no—
100C22×Q8×C61D411yescharacteristic
101C22×D4×C61C2311yescharacteristic
102C23×C4×C61—21no—
103C23×C4×C61—21no—
104C23×C4×C61—21no—
105C4⋊Q8×C61—22no—
106C4⋊Q8×C61—22no—
107#125×C61—22no—
108#125×C61—22no—
109#125×C61—22no—
110#125×C61D414yes—
111C22⋊Q8×C61—28no—
112C22⋊Q8×C61—28no—
113C22⋊Q8×C61—28no—
114C4×Q8×C61—22no—
115C4×Q8×C61—22no—
116C4×D4×C61—22no—
117C4×D4×C61—22no—
118C4×D4×C61—22no—
119C4×D4×C61—22no—
120C4×D4×C61—22no—
121C4×D4×C61D414yes—
122C2×C4⋊C4×C61—28no—
123C2×C4⋊C4×C61—21no—
124C2×C4⋊C4×C61—21no—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61—21no—
128C2×C4⋊C4×C61D411yescharacteristic
129C2×C4⋊C4×C61D411yescharacteristic
130C2×C4⋊C4×C61D411yescharacteristic
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C4⋊C4×C61C2311yescharacteristic
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C22⋊C4×C61—21no—
135C2×C22⋊C4×C61—21no—
136C2×C22⋊C4×C61—21no—
137C2×C42×C61C2311yescharacteristic
138C2×C42×C61C2311yescharacteristic
139C2×C42×C61C2311yescharacteristic
140C24×C61—21no—
141C2×Q8×C61—24no—
142C2×Q8×C61—22no—
143C2×Q8×C61—22no—
144C2×Q8×C61—41no—
145C2×Q8×C61—81no—
146C2×Q8×C61—81no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61C2×D412yes—
151C2×D4×C61C2×Q812yes—
152C2×D4×C61#4112yes—
153C22×C4×C61—24no—
154C22×C4×C61—21no—
155C22×C4×C61—21no—
156C22×C4×C61—21no—
157C22×C4×C61—24no—
158C22×C4×C61—24no—
159C22×C4×C61—42no—
160C22×C4×C61—42no—
161C22×C4×C61—81no—
162C22×C4×C61—81no—
163C22×C4×C61—81no—
164C22×C4×C61—81no—
165C22×C4×C61C2×D411yescharacteristic
166C22×C4×C61C2×D411yescharacteristic
167C22×C4×C61C2×D411yescharacteristic
168C22×C4×C61C2×D411yescharacteristic
169C22×C4×C61C2×D411yescharacteristic
170C22×C4×C61C2×D411yescharacteristic
171C22×C4×C61C2411yescharacteristic
172C22×Dic61—21no—
173C22×Dic61—21no—
174C22×Dic61—21no—
175C22×Dic61—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122—21no—
181C2×Dic122C2×Q812yes—
182C4⋊C4×C61—24no—
183C4⋊C4×C61—24no—
184C4⋊C4×C61—24no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61—24no—
187C4⋊C4×C61—24no—
188C4⋊C4×C61—22no—
189C4⋊C4×C61—22no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61C2×D412yes—
194C4⋊C4×C61C2×D412yes—
195C4⋊C4×C61C2×D412yes—
196C22⋊C4×C61—24no—
197C22⋊C4×C61—24no—
198C22⋊C4×C61—24no—
199C42×C61—21no—
200C42×C61—21no—
201C42×C61—21no—
202C42×C61—21no—
203C42×C61C2×D412yes—
204C42×C61C2×D412yes—
205#11758200—28no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—42no—
213C4⋊Dic61—81no—
214#11758196—42no—
215#11758196—81no—
216#11758196—81no—
217C4×Dic61—42no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418C2×Q812yes—
230C23×C61—22no—
231C23×C61—22no—
232C23×C61—22no—
233C23×C61—41no—
234C23×C61—41no—
235C23×C61—41no—
236C23×C61—41no—
237C23×C61C22×D411yescharacteristic
238Q8×C61—44no—
239Q8×C61—84no—
240D4×C61—22no—
241D4×C61—22no—
242D4×C61—22no—
243D4×C61C22⋊Q814yes—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—22no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61—21no—
261C2×C4×C61—21no—
262C2×C4×C61—48no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61—84no—
270C2×C4×C61—84no—
271C2×C4×C61#12012yes—
272C2×C4×C61#12012yes—
273C2×C4×C61#13712yes—
274C2×C4×C61C22×D411yescharacteristic
275C2×C4×C61C22×D411yescharacteristic
276C2×C4×C61C22×Q811yescharacteristic
277C2×C4×C61C2×#4111yescharacteristic
278C2×C4×C612−1+411yescharacteristic
279C2×C4×C612−1+411yescharacteristic
280C2×Dic61—22no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C2×Dic61—41no—
292Dic122—22no—
293Dic122—22no—
294Dic122—41no—
295Dic122—41no—
296#92418—22no—
297#92418—41no—
298#92418—41no—
299#92418—41no—
300#24349—611nomaximal, Sylow
301C22×C61—24no—
302C22×C61—42no—
303C22×C61—42no—
304C22×C61—42no—
305C22×C61—42no—
306C22×C61—42no—
307C22×C61—42no—
308C22×C61C2×#12011yescharacteristic
309C22×C61#54811yescharacteristic
310C22×C61D4×Q811yescharacteristic
311C22×C61C2×#13711yescharacteristic
312C22×C61#57511yescharacteristic
313C22×C61#57611yescharacteristic
314C22×C61#57711yescharacteristic
315C4×C61—22no—
316C4×C61—22no—
317C4×C61—22no—
318C4×C61—41no—
319C4×C61—42no—
320C4×C61—41no—
321C4×C61—41no—
322C4×C61—42no—
323C4×C61—42no—
324C4×C61C2×C22⋊Q811yescharacteristic
325C4×C61#53611yescharacteristic
326C4×C61#54811yescharacteristic
327C4×C61D4×Q811yescharacteristic
328Dic61—42no—
329Dic61—81no—
330Dic61—81no—
331Dic61—81no—
332C2×#554D6111yescharacteristic
333C2×#479—611no—
334C2×#440—611no—
335#2582—611no—
336#2059—611no—
337#2057—611no—
338#2026—611no—
339#2024—611no—
340#1992—611no—
341#1985—611no—
342#1877—611no—
343#1877—611no—
344#1864—611no—
345#1862—611no—
346#1802—611no—
347C2×C61—44no—
348C2×C61#240911yescharacteristic
349C2×C61#300711yescharacteristic
350C2×C61#301211yescharacteristic
351C2×C61#337011yescharacteristic
352C2×C61#337211yescharacteristic
353C2×C61#338511yescharacteristic
354C2×C61#338811yescharacteristic
355C22×SD16—611no—
356#554—24no—
357#554—24no—
358C2×C4⋊Q8C2×D6111yescharacteristic
359C2×#125C2×D6111yescharacteristic
360C2×#125C2×D6111yescharacteristic
361C2×C22⋊Q8—21no—
362C2×C22⋊Q8—21no—
363C2×C22⋊Q8—21no—
364C2×C4×Q8—611no—
365C2×C4×Q8C2×D6111yescharacteristic
366C2×C4×D4C2×D6111yescharacteristic
367C2×C4×D4C2×D6111yescharacteristic
368C2×C4×D4C2×D6111yescharacteristic
369C22×C4⋊C4—21no—
370#479—1224no—
371#440—1224no—
372C2×#107—611no—
373C2×#107—611no—
374C2×#106—611no—
375C2×C4⋊C8—611no—
376C2×#103—611no—
377C2×#103—611no—
378C2×#103—611no—
379C2×#102—611no—
380C2×#102—611no—
381C2×#102—611no—
382C2×#97—611no—
383#398—1221no—
384#398—1221no—
385#391—611no—
386#391—611no—
387#391—1221no—
388#389—611no—
389#389—611no—
390#389—1221no—
391#384—611no—
392#340—611no—
393#340—611no—
394#340—611no—
395#340—611no—
396#340—611no—
397#340—611no—
398#336—611no—
399#336—611no—
400#336—611no—
401#336—611no—
402C61#2434911yescharacteristic, complemented, Sylow
403C22×Q8—21no—
404C22×Q8—611no—
405C22×Q8C22⋊D6111yescharacteristic
406C22×D4C22×D6111yescharacteristic
407C23×C4—21no—
408C23×C4—21no—
409C23×C4—21no—
410C2×SD16—614no—
411C2×SD16—1222no—
412C2×SD16—1222no—
413C22×C8—611no—
414C22×C8—611no—
415C22×C8—611no—
416C22×C8—611no—
417C4⋊Q8—22no—
418C4⋊Q8—22no—
419#125—22no—
420#125—22no—
421#125—22no—
422#125C22⋊D6114yes—
423C22⋊Q8—28no—
424C22⋊Q8—28no—
425C22⋊Q8—28no—
426C4×Q8—22no—
427C4×Q8—22no—
428C4×Q8—1222no—
429C4×Q8—1222no—
430C4×D4—22no—
431C4×D4—22no—
432C4×D4—22no—
433C4×D4—22no—
434C4×D4—22no—
435C4×D4C22⋊D6114yes—
436C2×C4⋊C4—28no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—21no—
441C2×C4⋊C4—21no—
442C2×C4⋊C4—611no—
443C2×C4⋊C4—611no—
444C2×C4⋊C4—611no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4—1221no—
450C2×C4⋊C4C22⋊D6111yescharacteristic
451C2×C4⋊C4C22⋊D6111yescharacteristic
452C2×C4⋊C4C22⋊D6111yescharacteristic
453C2×C4⋊C4C22×D6111yescharacteristic
454C2×C4⋊C4C22×D6111yescharacteristic
455C2×C4⋊C4C22×D6111yescharacteristic
456C2×C22⋊C4—21no—
457C2×C22⋊C4—21no—
458C2×C22⋊C4—21no—
459C2×C42—1221no—
460C2×C42C22×D6111yescharacteristic
461C2×C42C22×D6111yescharacteristic
462C2×C42C22×D6111yescharacteristic
463#107—1222no—
464#107—1222no—
465#106—1222no—
466C4⋊C8—1222no—
467#103—1222no—
468#103—1222no—
469#103—1222no—
470#102—1222no—
471#102—1222no—
472#102—1222no—
473#97—1222no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478#95—1221no—
479#95—1221no—
480C24—21no—
481C2×Q8—24no—
482C2×Q8—22no—
483C2×Q8—22no—
484C2×Q8—41no—
485C2×Q8—81no—
486C2×Q8—81no—
487C2×Q8—612no—
488C2×Q8—1221no—
489C2×Q8—1221no—
490C2×Q8—1221no—
491C2×Q8—1221no—
492C2×Q8—1221no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4Q8×D6112yes—
497C2×D4#1175822612yes—
498C2×D4C2×C22⋊D6112yes—
499C22×C4—24no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—24no—
504C22×C4—24no—
505C22×C4—42no—
506C22×C4—42no—
507C22×C4—81no—
508C22×C4—81no—
509C22×C4—81no—
510C22×C4—81no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4C2×C22⋊D6111yescharacteristic
516C22×C4C2×C22⋊D6111yescharacteristic
517C22×C4C2×C22⋊D6111yescharacteristic
518C22×C4C2×C22⋊D6111yescharacteristic
519C22×C4C2×C22⋊D6111yescharacteristic
520C22×C4C2×C22⋊D6111yescharacteristic
521C22×C4C23×D6111yescharacteristic
522SD16—1228no—
523C2×C8—612no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—1221no—
531C2×C8—1221no—
532C2×C8—1221no—
533C2×C8—1221no—
534C2×C8—1221no—
535C4⋊C4—24no—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—22no—
542C4⋊C4—24no—
543C4⋊C4—24no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—1221no—
552C4⋊C4—4881no—
553C4⋊C4—4881no—
554C4⋊C4—4881no—
555C4⋊C4—2442no—
556C4⋊C4—2442no—
557C4⋊C4C2×C22⋊D6112yes—
558C4⋊C4C2×C22⋊D6112yes—
559C4⋊C4C2×C22⋊D6112yes—
560C22⋊C4—24no—
561C22⋊C4—24no—
562C22⋊C4—24no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—21no—
567C42—2442no—
568C42C2×C22⋊D6112yes—
569C42C2×C22⋊D6112yes—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23C22×C22⋊D6111yescharacteristic
578Q8—44no—
579Q8—84no—
580Q8—1222no—
581Q8—1222no—
582Q8—2441no—
583Q8—2441no—
584D4—22no—
585D4—22no—
586D4—22no—
587D4#1451975314yes—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—48no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—1222no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4#1451974612yes—
628C2×C4#1451974612yes—
629C2×C4#1451980712yes—
630C2×C4C22×C22⋊D6111yescharacteristic
631C2×C4C22×C22⋊D6111yescharacteristic
632C2×C4C2×Q8×D6111yescharacteristic
633C2×C4C2×#1175822611yescharacteristic
634C2×C4#1451982511yescharacteristic
635C2×C4#1451982511yescharacteristic
636C8—1222no—
637C8—2441no—
638C8—2441no—
639C8—2441no—
640C22—24no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22C2×#1451974611yescharacteristic
648C22#1684042311yescharacteristic
649C22Q8×C22⋊D6111yescharacteristic
650C22C2×#1451980711yescharacteristic
651C22#1684056011yescharacteristic
652C22#1684056511yescharacteristic
653C22#1684057311yescharacteristic
654C4—22no—
655C4—22no—
656C4—22no—
657C4—41no—
658C4—41no—
659C4—42no—
660C4—42no—
661C4—41no—
662C4—42no—
663C4—4881no—
664C4—4881no—
665C4—4881no—
666C4—2442no—
667C4C2×#1451975311yescharacteristic
668C4#1684033611yescharacteristic
669C4#1684042311yescharacteristic
670C4Q8×C22⋊D6111yescharacteristic
671C2—44no—
672C2#2144917811yescharacteristic
673C2#2145507911yescharacteristic
674C2#2145512311yescharacteristic
675C2#2145857211yescharacteristic
676C2#2145860011yescharacteristic
677C2#2145872911yescharacteristic
678C2#2145876411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#554C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenterC2×#554D61
hypocenterC61#24349
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#554

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16840066C2semidirect
3. #21451189C2semidirect
4. #21447154C2semidirect
5. #21447144C2semidirect
6. #21446967C2semidirect
7. #21446957C2semidirect
8. #21446789C2semidirect
9. #21446760C2semidirect
66. C2×#11758200Q8semidirect
402. C61#24349semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168400662
C2×#168398622
#214511892
#214471542
#214471442
#214469672
#214469572
#214467892
#214467602
#214461532
#214460802
#214460662
#214458192
C2×C61×#5542
#2434961